Definition:Local Minimum

Definition

Let $f$ be a real function defined on an open interval $\left({a .. b}\right)$.

Let $\xi \in \left({a .. b}\right)$.

Then $f$ has a local minimum at $\xi$ iff:

$\exists \left({c .. d}\right) \subseteq \left({a .. b}\right): \forall x \in \left({c .. d}\right): f \left({x}\right) \ge f \left({\xi}\right)$.

That is, iff there is some subinterval on which $f$ attains a minimum within that interval.

Notes

Note the requirement for the intervals to be open. A closed interval of course includes the value of $f$ at its end points and so every closed interval attains a minimum.

Also see

• Local maximum

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 11.1$